4.2 Mass transfer

Although there are still many unanswered theoretical questions about the nature of the mass
transfer phase, the basic properties of the evolution of a binary due to mass transfer can easily
be described. The rate at which a star can adjust to changes in its mass is governed by three
time scales. The dynamical time scale results from the adiabatic response of the star to restore
hydrostatic equilibrium, and can be approximated by the free fall time across the radius of the star,

where and are the mass and radius of the star. The thermal equilibrium of the star is restored
over a longer period given by the thermal time scale

where is the luminosity of the star. Finally, the main-sequence lifetime of the star itself provides a third
time scale, which is also known as the nuclear time scale:

The rate of mass transfer/loss from the Roche lobe filling star is governed by how the star’s radius
changes in response to changes in its mass. Hjellming and Webbink [108] describe these changes and the
response of the Roche lobe to mass changes in the binary using the radius-mass exponents,
, for each of the three processes described in Equations (13, 14, 15) and defining

for the Roche lobe radius-mass exponent. If , the star cannot adjust to the Roche lobe, then the
mass transfer takes place on a dynamical time scale and is limited only by the rate at which material can
stream through the inner Lagrange point. If , then the mass transfer rate is governed by
the slow expansion of the star as it relaxes toward thermal equilibrium, and it occurs on a thermal
time scale. If both and are greater than , then the mass loss is driven either
by stellar evolution processes or by the gradual shrinkage of the orbit due to the emission of
gravitational radiation. The time scale for both of these processes is comparable to the nuclear
time scale. A good analysis of mass transfer in cataclysmic variables can be found in King et
al. [130].

Conservative mass transfer occurs when there is no mass loss from the system. During conservative mass
transfer, the orbital elements of the binary can change. Consider a system with total mass
and semi-major axis . The total orbital angular momentum

is a constant, and we can write . Using Kepler’s third law and denoting the initial values by
a subscript , we find:

Note that if the more massive star loses mass, then the orbital period decreases and the orbit shrinks. If the
less massive star is the donor, then the orbit expands. Usually, the initial phase of RLOF takes place as the
more massive star evolves. As a consequence, the orbit of the binary will shrink, driving the binary to a
more compact orbit.

In non-conservative mass transfer, both mass and angular momentum can be removed from the system.
There are two basic non-conservative processes which are important to the formation of relativistic
binaries – the common-envelope process and the supernova explosion of one component of the
binary. The result of the first process is often a short-period, circularized binary containing a
white dwarf. Although the most common outcome of the second process is the disruption of
the binary, occasionally this process will result in an eccentric binary containing a neutron
star.

Common envelope scenarios result when one component of the binary expands so rapidly that the mass
transfer is unstable and the companion becomes engulfed by the donor star. The companion then ejects the
envelope of the donor star. The energy required to eject the envelope comes from the orbital energy
of the binary and thus the orbit shrinks. The efficiency of this process determines the final
orbital period after the common envelope phase. This is described by the efficiency parameter

where is the binding energy of the mass stripped from the envelope and is the change in
the orbital energy of the binary. The result of the process is the exposed degenerate core of the
donor star in a tight, circular orbit with the companion. This process can result in a double
degenerate binary if the process is repeated twice or if the companion has already evolved to
a white dwarf through some other process. A brief description of the process is outlined by
Webbink [238], and a discussion of the factors involved in determining is presented in Sandquist et
al. [206].

The effect on a binary of mass loss due to a supernova can be quite drastic. Following Padmanabhan [172],
this process is outlined using the example of a binary in a circular orbit with radius . Let be the
velocity of one component of the binary relative to the other component. The initial energy of the binary is
given by

Following the supernova explosion of , the expanding mass shell will quickly cross the orbit of ,
decreasing the gravitational force acting on the secondary. The new energy of the binary is then

where is the mass of the remnant neutron star. We have assumed here that the passage of the mass
shell by the secondary has negligible effect on its velocity (a safe assumption, see Pfahl et al. [177] for a
discussion), and that the primary has received no kick from the supernova (not necessarily a
safe assumption, but see Davies and Hansen [46] or Pfahl et al. [178] for an application to
globular cluster binaries). Since we have assumed that the instantaneous velocities of both
components have not been affected, we can replace them by , and so

Note that the final energy will be positive and the binary will be disrupted if .
This condition occurs when the mass ejected from the system is greater than half of the initial total mass,

where . If the binary is not disrupted, the new orbit becomes eccentric and expands to a
new semi-major axis given by

and orbital period

We have seen that conservative mass transfer can result in a tighter binary if the more massive star is
the donor. Non-conservative mass transfer can also drive the components of a binary together during a
common envelope phase when mass and angular momentum are lost from the system. Direct mass loss
through a supernova explosion can also alter the properties of a binary, but this process generally drives the
system toward larger orbital separation and can disrupt the binary entirely. With this exception, the
important result of all of these processes is the generation of tight binaries with at least one degenerate
object.

The processes discussed so far apply to the generation of relativistic binaries anywhere. They occur
whenever the orbital separation of a progenitor binary is sufficiently small to allow for mass transfer or
common envelope evolution. Population distributions for relativistic binaries are derived from an initial
mass function, a distribution in mass ratios, and a distribution in binary separations. These
initial distributions are then fed into models for binary evolution such as StarTrack [18] or
SeBa [191, 168] in order to determine rates of production of relativistic binaries. The evolution of the
binary is often determined by the application of some simple operational formulae such as those
described by Tout et al. [229] or Hurley et al. [111]. For example, Hils, Bender, and Webbink [107]
estimated a population of close white dwarf binaries in the disk of the galaxy using a Salpeter
mass function, a mass ratio distribution strongly peaked at 1, and a separation distribution
that was flat in . Other estimates of relativistic binaries differ mostly by using different
distributions [17, 119, 168, 167].